Deriving Equations for Light Sphere in Collinear Motion - O and O' Observers

[cfrogue]

Note that light striking the left endpoint of the primed rod is a different event from light striking the right endpoint of the primed rod.

For O, light hits the left endpoint of the unprimed rod at the same t as light hits the right endpoint of the unprimed rod.

For O, light hits the left endpoint of the primed rod at an earlier t than light hits the right endpoint of the primed rod.

For O', light hits the left endpoint of the primed rod at the same t' as light hits the right endpoint of the primed rod.

For O', light hits the left endpoint of the unprimed rod at a later t' than light hits the right endpoint of the unprimed rod.

I completely agree.

Now, when in the coords of O are the points of O' struck at the same time.

That is the question.

If you come up with two answers for O, the O' will see the strikes at the same time twice.

 

[atyy]

I proceed by reductio ad absurdum there exists only time t in O for this t'.

Assume there exists a tx < t such that the points in O' are struck at the same time.
Then tx < r/(λ(c-v))
tx = (t' + vx'/c^2)λ < r/(λ(c-v))
We have by the SR spherical light sphere,
ct' = x', t' = x'/c
Also, by selection x' = r.
( x'/c + vx'/c^2)λ < r/(λ(c-v))
(r/c + rv/c^2)λ < r/(λ(c-v))
(1/c + 1v/c^2)λ < 1/(λ(c-v))
((c + v)/c^2)λ < 1/(λ(c-v))
(c + v) < c^2/(λ^2(c-v))
(c + v) < (c^2/(c-v))((c^2 - v^2)/c^2)
c + v < (c^2 - v^2)/(c - v)
c + v < c + v
0 < 0

This is a contradiction. The same argument hold for tx > t.
Thus, the calculated t is the unique time in O when the points of O' are struck at the same time.

BTW, this looks correct, except for the final sentence "Thus, the calculated t is the unique time in O when the points of O' are struck at the same time." So what we are having is a problem in interpreting the mathematics.

I do not think so since I showed no other t fits the bill > t or < t.

The final sentence is wrong because "the points of O' are struck at the same time" refers to 2 events. However, by the restriction x'=r, you can only consider the event of light hitting the right endpoint of the primed rod. You cannot consider the the event of light hitting the left endpoint of the primed rod, at which x'=-r. So the reference to 2 events in the final sentence is not justified.

 

[Al68]

I completely agree.

Now, when in the coords of O are the points of O' struck at the same time.

That is the question.

If you come up with two answers for O, the O' will see the strikes at the same time twice.

You mean the way you did exactly that here:

O' sees the strikes as simultaneous whereas O sees them at different times.

I don't know why you keep contradicting yourself, unless there is a major underlying issue that is a mystery to me.

 

[cfrogue]

Are you referring to a time at which an observer physically sees light reflected back from the rod's endpoints?

No, I am only applying LT.
I do not use the reflected logic.

That is when my eyes will see it. I want to know when it actually happens which LT gives.

Do you think there should be two times in O when the points of O' are struck at the same time?

According to SR, there are. And you say so next:

SR does not say O' will see simultaneous strikes at two different times. That contradicts reality.

Can you prove this?

Do you not realize that means that t(L) is a different value than t(R), meaning more than one t in O for the single t' in O' [t'=t'(L)=t'(R)]?

That is false and you do not understand the mapping of LT.

There is more than one t in O, but only one in O'. That is LT.

Do you understand that given x^2 = 9, that -3 and 3 fit the bill?

This is the way LT works and we are exploring this.

This is R of S combined with the light postulate in O'.

O will see the strikes at different times but O' will see them at the same time.

 

[atyy]

Now, when in the coords of O are the points of O' struck at the same time.

That is the question.

According to the relativity of simultaneity, this question is meaningless.

 

[cfrogue]

The final sentence is wrong because "the points of O' are struck at the same time" refers to 2 events. However, by the restriction x'=r, you can only consider the event of light hitting the right endpoint of the primed rod. You cannot consider the the event of light hitting the left endpoint of the primed rod, at which x'=-r. So the reference to 2 events in the final sentence is not justified.

Not correct.

LT provides for two different times to be mapped to one time.

That is R of S and the light postulate O'.

So, O sees two different times for L' and R' of O' but O' sees them at the same time.

Where is this wrong?

 

[cfrogue]

According to the relativity of simultaneity, this question is meaningless.

This is not true.

This is a function.

given x^2 and -3 and 3, both map to the same value in the other system.

Where is the problem?

 

[cfrogue]

According to the relativity of simultaneity, this question is meaningless.

I need to say this.

There is no question O' sees its points hit at the same time by the light postulate.

If LT cannot handle this, according to your logic, then SR cannot map its own logic correctly.

I have shown it does work.

 

[Al68]

SR does not say O' will see simultaneous strikes at two different times. That contradicts reality.

Can you prove this?

Why would I try to prove something I never said?

Do you not realize that means that t(L) is a different value than t(R), meaning more than one t in O for the single t' in O' [t'=t'(L)=t'(R)]?

That is false and you do not understand the mapping of LT.

If that is false, then why do you say exactly that next:

There is more than one t in O, but only one in O'. That is LT.

I agree. Why wouldn't I agree with a statement that I made, which you claimed to be false.

 

[atyy]

I have shown it does work.

The step x'=r in your derivation is not justified if you want to be able to refer to both events in the final interpretation. Once you restrict yourself to x'=r, all successive steps can only refer to events that occur at x'=r, not at x'=-r.

 

[cfrogue]

If that is false, then why do you say exactly that next:I agree. Why wouldn't I agree with a statement that I made, which you claimed to be false.

I said:

There is more than one t in O, but only one in O'. That is LT.

I am simply not getting the problem.

I said before, -3, 3 under x^2 = 9.

What is the problem?

There are two different values in O, for example, -3 and 3, and one value in O', 9.

LT is consistent.

 

[cfrogue]

The step x'=r in your derivation is not justified if you want to be able to refer to both events in the final interpretation. Once you restrict yourself to x'=r, all successive steps can only refer to events that occur at x'=r, not at x'=-r.

Nope, I was careful to operate with r in the x' system.

I never crossed over in the O system with this r without r/λ.

Check the logic.

 

[Al68]

I said:


I am simply not getting the problem.

I said before, -3, 3 under x^2 = 9.

What is the problem?

There are two different values in O, for example, -3 and 3, and one value in O', 9.

LT is consistent.

That's exactly what others have been telling you for hundreds of posts while you (and only you) were saying things that contradict it like:

Thus, the calculated t is the unique time in O when the points of O' are struck at the same time.

 

[atyy]

Nope, I was careful to operate with r in the x' system.

I never crossed over in the O system with this r without r/λ.

Check the logic.

The problem is not crossing over into O. The problem is that the light sphere is x'=r and x'=-r, and in your final sentence you refer to events at x'=r and x'=-r, yet in the middle you have excluded all points x'=-r.

 

[cfrogue]

That's exactly what others have been telling you for hundreds of posts while you (and only you) were saying things that contradict it like:

Yes, I am now certain I know you.

Can you prove your claim so I can learn?

 

[cfrogue]

The problem is not crossing over into O. The problem is that the light sphere is x'=r and x'=-r, and in your final sentence you refer to events at x'=r and x'=-r, yet in the middle you have excluded all points x'=-r.

No,the light postulate says

ct' = ±x'

I included all this in my equations.

Do you have a math equation that is different?

Can I see it?

 

[Al68]

Yes, I am now certain I know you.

Who am I then?

Can you prove your claim so I can learn?

Which claim?

 

[atyy]

No,the light postulate says

ct' = ±x'

I included all this in my equations.

Do you have a math equation that is different?

Can I see it?

Hence x'=-ct' and x'=+ct'

 

[cfrogue]

Hence x'=-ct' and x'=+ct'

this is what I used.

Note simultaneity occurs when x' = ct'.

All of it falls in place after that.

Did you say my equations are false?

Can you show me the two different times in O' when it sees simultaneity.

I would like to see the math.

 

[cfrogue]

Who am I then?Which claim?

Just prove my math is false and then I can learn.

 

[atyy]

this is what I used.

Note simultaneity occurs when x' = ct'.

All of it falls in place after that.

Did you say my equations are false?

Can you show me the two different times in O' when it sees simultaneity.

I would like to see the math.

Yes, there is one time in O' when it sees simultaneity, but there are two locations x'=ct' and x'=-ct'. In your derivation, you restrict x'=r, so you restrict to one location.

 

[cfrogue]

Yes, there is one time in O' when it sees simultaneity, but there are two locations x'=ct' and x'=-ct'. In your derivation, you restrict x'=r, so you restrict to one location.

There is nothing wrong with this.

I must restrict it to one time location in O' by the light postulate.

It says, x'=ct' and x'=-ct', so I must follow the rules. There is one time for simultaneity in O'.

Do you see this?

 

[atyy]

There is nothing wrong with this.

I must restrict it to one time location in O' by the light postulate.

It says, x'=ct' and x'=-ct', so I must follow the rules. There is one time for simultaneity in O'.

Do you see this?

Yes, one t' coordinate, but two x' coordinates. You excluded one x' coordinate when you used x'=r in your derivation.

 

[Al68]

Just prove my math is false and then I can learn.

Math isn't true or false, claims are. Your various claims contradict each other, therefore some of them are false.

Note simultaneity occurs when x' = ct'.

This makes no sense. Simultaneity isn't an event, it's a description of multiple events.

 

[cfrogue]

Yes, one t' coordinate, but two x' coordinates. You excluded one x' coordinate when you used x'=r in your derivation.

I used the light postulate to conclude in O' x +- ct' are simultaneous.



inxs - elegantly wasted

 

[cfrogue]

Math isn't true or false, claims are. Your various claims contradict each other, therefore some of them are false.This makes no sense. Simultaneity isn't an event, it's a description of multiple events.

Are you saying this is true in O'?

Can you prove this?

O' sees all its points struck at the same time.

 

[cfrogue]

Math isn't true or false, claims are. Your various claims contradict each other, therefore some of them are false.

Looks like I am wrong.

Can you show me?

 

[Al68]

Are you saying this is true in O'?

Can you prove this?

O' sees all its points struck at the same time.

What are you talking about? O' doesn't see all of its points struck at the same time, only any two points equally distance from the origin.

 

[Al68]

Looks like I am wrong.

Can you show me?

I, and others better than I, have tried repeatedly, and still haven't given up yet.

But I would just read Einstein's 1905 paper for this. It's not the only source out there, and maybe not the best, but it's certainly more than good enough for this topic.

 

[cfrogue]

What are you talking about? O' doesn't see all of its points struck at the same time, only any two points equally distance from the origin.

Well, actually if O' were a rigid body sphere, all points are struck at the same time, the same as O.

So, yes, all the points of O' are struck at the same time.

Say, do you have the math to refute the light postulate in O'?

 

[cfrogue]

I, and others better than I, have tried repeatedly, but still haven't given up yet.

But I would just read Einstein's 1905 paper for this. It's not the only source out there, and maybe not the best, but it's certainly more than good enough for this topic.

Yes, may I see the math please?

 

[Al68]

Well, actually if O' were a rigid body sphere, all points are struck at the same time, the same as O.

So, yes, all the points of O' are struck at the same time.

Say, do you have the math to refute the light postulate in O'?

There's a big difference in saying that all points in O' are struck at the same time and saying only that all points on a sphere in O' are struck at the same time.

The light postulate says all points on the sphere in O', not all points in O', would be hit at the same time in O'.

I have to assume that's what you really meant.

 

[cfrogue]

There's a big difference in saying that all points in O' are struck at the same time and saying only that all points on a sphere in O' are struck at the same time.

The light postulate says all points on the sphere in O', not all points in O', would be hit at the same time in O'.

I have to assume that's what you really meant.

Yes, that is what I meant. Just the surface of the sphere would O' see see simultaneity.

Thanks

 

[Al68]

Yes, may I see the math please?

Sure. What specifically do you want to see the math for?

 

[cfrogue]

Sure. What specifically do you want to see the math for?

Well, you said there exists two points in O that are simultaneous in O'

Can I see this?

 

[atyy]

I used the light postulate to conclude in O' x +- ct' are simultaneous.
inxs - elegantly wasted

Thanks for the music! Yes, x'= +-ct' means for one t' there are two x's. You omitted the +- in your derivation below. In the final sentence there you refer to "points of O' are struck at the same time". However, by omitting the +- in your derivation, you are excluding one point. Since there are only two points on the x' axis for fixed t', only one point remains. So the plural "points" is not justified.

I proceed by reductio ad absurdum there exists only time t in O for this t'.

Assume there exists a tx < t such that the points in O' are struck at the same time.
Then tx < r/(λ(c-v))
tx = (t' + vx'/c^2)λ < r/(λ(c-v))
We have by the SR spherical light sphere,
ct' = x', t' = x'/c
Also, by selection x' = r.
( x'/c + vx'/c^2)λ < r/(λ(c-v))
(r/c + rv/c^2)λ < r/(λ(c-v))
(1/c + 1v/c^2)λ < 1/(λ(c-v))
((c + v)/c^2)λ < 1/(λ(c-v))
(c + v) < c^2/(λ^2(c-v))
(c + v) < (c^2/(c-v))((c^2 - v^2)/c^2)
c + v < (c^2 - v^2)/(c - v)
c + v < c + v
0 < 0

This is a contradiction. The same argument hold for tx > t.
Thus, the calculated t is the unique time in O when the points of O' are struck at the same time.

 

[Al68]

Well, you said there exists two point in O that are simultaneous in O'

Can I see this?

I never said that. "Points" can't be simultaneous, events can be. I said that two events simultaneous in O' (light reaching either rod end) occur at two different times in O.

For that single t' in O', t = gamma(t' - vx'/c^2). Since there are two different values for x', there will be two different values for t.

 

[cfrogue]

Thanks for the music! Yes, x'= +-ct' means for one t' there are two x's. You omitted the +- in your derivation below. In the final sentence there you refer to "points of O' are struck at the same time". However, by omitting the +- in your derivation, you are excluding one point. Since there are only two points on the x' axis for fixed t', only one point remains. So the plural "points" is not justified.

I do not need to include "You omitted the +- in your derivation below" +-, because the light postulate says the points are simultaneous.

This is axiomatic and I do not need to prove it.

So, given the simultaneity by the light postulate, I need only consider one point.

 

[cfrogue]

I never said that. "Points" can't be simultaneous, events can be. I said that two events simultaneous in O' (light reaching either rod end) occur at two different times in O.

For that single t' in O', t = gamma(t' - vx'/c^2). Since there are two different values for x', there will be two different values for t.

It is OK to have two different values for t.



It is not OK to have two different values for the simultaneity of O' ie, two different t'.

You said it was.

Do you have the proof?

 

[Al68]

I do not need to include "You omitted the +- in your derivation below" +-, because the light postulate says the points are simultaneous.

This is axiomatic and I do not need to prove it.

So, given simultaneity by the light postulate, I need only consider one point.

That's not true. The light postulate only says the events are simultaneous in O'. They are still two separate events that correspond to two different locations in O' and therefore two different times in O.

 

[Al68]

It is not OK to have two different values for the simultaneity of O' ie, two different t'.

You said it was.

Do you have the proof?

When did I say that?

 

[atyy]

I do not need to include "You omitted the +- in your derivation below" +-, because the light postulate says the points are simultaneous.

This is axiomatic and I do not need to prove it.

So, given the simultaneity by the light postulate, I need only consider one point.

If what you say is true, then you should be able to obtain the same result with x'=-r.

 

[Dale]

Looks like I missed a lot last night. I don't know if this was already resolved, but cfrogue's proof is incorrect. Specifically:

We have by the SR spherical light sphere,
ct' = x', t' = x'/c
Also, by selection x' = r.

These two lines are wrong. It should be:
"We have by the SR spherical light sphere,
ct' = ±x', t' = ±x'/c
Also, by selection ±x' = r."

The fact that there are two times in the unprimed frame follows from that.

 

[cfrogue]

If what you say is true, then you should be able to obtain the same result with x'=-r.

This is curious.

When x'=-r in O', that is at the time r/(λ(c+v)) in O.
When x'=+r in O', that is at the time r/(λ(c-v)) in O.

Thus, two different times in O are producing simultaneity in O'.

I'm thinking about all this.

What is your view at this point?

 

[cfrogue]

Looks like I missed a lot last night. I don't know if this was already resolved, but cfrogue's proof is incorrect. Specifically:These two lines are wrong. It should be:
"We have by the SR spherical light sphere,
ct' = ±x', t' = ±x'/c
Also, by selection ±x' = r."

The fact that there are two times in the unprimed frame follows from that.

Yes, I agree and just caught that also at the same time you posted.

 

[cfrogue]

Looks like I missed a lot last night. I don't know if this was already resolved, but cfrogue's proof is incorrect. Specifically:These two lines are wrong. It should be:
"We have by the SR spherical light sphere,
ct' = ±x', t' = ±x'/c
Also, by selection ±x' = r."

The fact that there are two times in the unprimed frame follows from that.

This seems strange.

As t starts at zero in O and advances to t1 = r/(λ(c+v)), ct' < |x'|.
At that point in the time of O, ct' = -x'.

Then the negative x' continues to go more negative as time increases in O.

Then when the time in O reaches, t2 = r/(λ(c-v)), ct' = +x'.

Is this correct?

 

[Dale]

Yes, I agree and just caught that also at the same time you posted.

No problem. I have updated my spacetime diagram to highlight the events where the light cone strikes the ends of the rods. The green dots are the events where the light cone strikes the ends of the unprimed rod. The red dots are the events where the light cone strikes the ends of the primed rod.

 

[cfrogue]

No problem. I have updated my spacetime diagram to highlight the events where the light cone strikes the ends of the rods. The green dots are the events where the light cone strikes the ends of the unprimed rod. The red dots are the events where the light cone strikes the ends of the primed rod.

You do produce some excellent graphics.

 

[cfrogue]

No problem. I have updated my spacetime diagram to highlight the events where the light cone strikes the ends of the rods. The green dots are the events where the light cone strikes the ends of the unprimed rod. The red dots are the events where the light cone strikes the ends of the primed rod.

This implies there are two times in O t1, t2, such that the light stikes of the endpoints of the rod of O' are simultaneous in O' and t1 < t2.

Is this correct?

 

[atyy]

This is curious.

When x'=-r in O', that is at the time r/(λ(c+v)) in O.
When x'=+r in O', that is at the time r/(λ(c-v)) in O.

Thus, two different times in O are producing simultaneity in O'.

I'm thinking about all this.

What is your view at this point?

My view is that you are calculating the same thing as you did here, which was correct, and you had no problem interpreting:

Let's see if I understand R of S.

O sees the strikes of O' at
t_L = d/(2cλ(c+v))
t_R = d/(2cλ(c-v))

t_L < t_R
Is this R of S?

[Edited for correction]

[t_L = d/(2λ(c+v))
t_R = d/(2λ(c-v))]